Summary. As a toy model for the algebraic approach to the construction of , we give a brief introduction to generators, relations and syzygies in a group-theoretical context, and their use in the construction of group representations.
As we have already mentioned, there are many approaches to the construction of an invariant satisfying the condition in Equation (2). The algebraic approach, which is the topic of this article, is to find some algebraic context within which the set (or some mild generalization thereof) is finitely presented, and then to use this finite presentation to define . Namely, one would have to make wise guesses for the values of on the generators of , so that for each relation the corresponding values of would satisfy the corresponding relation (two comments: 1. For this to make sense must carry the same kind of algebraic structure as ; 2. The verification of essentiality, Equation (2), is typically easy).
Let us see what this entails on a toy model. Suppose we want to find invariants of elements of the set of braids on 4 strands. One way to proceed is to notice that carries an algebraic structure, that is, it has an associative product which makes it a group. Thus we may seek invariants on with values in associative algebras, which respect the algebraic structure. Such creatures are not new on the mathematical scenery; they are usually called ``group representations''. Our approach to finding representations of would be to make wise guesses for their values , and on the generators , and of (see Figure 1), so as to satisfy the relations between the 's. Setting , these relations are (again see Figure 1):
In our real problem, the construction of , the target space is graded, and we will attempt to construct inductively, degree by degree. Thus we will be asking ourselves, ``suppose our construction is done to degree 16; can we extend it to degree 17?''. Let us go back to the toy model and examine the situation over there. Let be an associative algebra and let be ideals in (think `` degrees and degrees'') so that (``''). Suppose we have which satisfy the equations (3) in (``done to degree 16''). But equations (3) may fail in ; let be the errors in when these equations are considered in :
We wish to modify the 's so as to satisfy equations (3) in (``extend to degree 17''), so we set
These are linear equations, and thus to solve our problem, namely to find 's so that , we need to show that the triple is in the image of the linear map defined by
Our strategy to show that is to find a second linear map , whose domain is the target space of , so that and so that . This done we can define the homology group , and if by some magical means we could prove that it vanishes, we would use to determine that , and our problem would be solved. We will mention techniques for the computation of the homology group in Section 5. For now we only wish to describe how the map is found.
To find linear relations between the errors , and , we start with a syzygy for our presentation of the braid group -- a closed loop whose vertices are words in the generators and whose edges are relations. When we perform the replacement on the vertices of a syzygy, say the one displayed in Figure 1, we get a loop like such:
Moral. It would be nice to have an algebraic context within which knot theory is finitely presented and within which the syzygies of the presentation are simple to analyze.